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Archaeology and geostatistics
C.D. Lloyda*, P.M. Atkinsonb
aSchool of Geography, Queen’s University, Belfast BT7 1NN,
UKbSchool of Geography, University of Southampton, Highfield,
Southampton SO17 1BJ, UK
Received 16 December 2002; received in revised form 24 June
2003; accepted 8 July 2003
Abstract
Geostatistics is used in many different disciplines to
characterise spatial variation and for spatial prediction, spatial
simulationand sampling design. Archaeology is an inherently spatial
discipline and the models and tools provided by geostatistics
should beas valuable in archaeology as they are in other
disciplines that are concerned with spatially varying properties.
However, therehave, so far, been few applications of geostatistics
in archaeology. This paper seeks to highlight some of the key tools
providedby geostatistics and to show, through two case studies, how
they may be employed in archaeological applications. Some
relevantliterature is summarised and two case studies are presented
based on the analysis of (i) Roman pottery and (ii) soil
phosphatedata.� 2003 Elsevier Ltd. All rights reserved.
Keywords: Spatial analysis; Mapping; Sampling design
1. Introduction
Geostatistics is a set of tools used for characterisingspatial
variation, spatial prediction, spatial simula-tion and spatial
optimisation (e.g., sampling design).Applications of geostatistics
are found in a wide range offields including biology, environmental
science, geogra-phy, geology, meteorology and mining. Geostatistics
isbased on the principle of spatial dependence (or
spatialautocorrelation): observations close in space tend to bemore
similar than those further apart. Therefore, if thespatial
distribution of some variable is structured (asopposed to being
random) geostatistics may be useful insome capacity.
The characterisation of spatial autocorrelation inarchaeological
variables has been the concern of severalresearchers. Hodder and
Orton [16], in their classic texton spatial analysis in
archaeology, provide a sectionon the subject of spatial
autocorrelation. This workincluded the definition of Moran’s I and
Geary’s c, two
coefficients which characterise the degree of spatial
auto-correlation in a variable. An application was demon-strated
based on the distribution of the length/breadthratio index of
Bronze Age spearheads. Specifically, I wasestimated for several
spatial lags (that is, for pairs oflocations separated by several
distance and directionvectors), enabling assessment of structure in
the spatialdistribution of the index. Other studies have
appliedsimilar statistical measures of spatial autocorrelationto
the terminal distribution of dated monuments atlowland Maya sites
[20,40].
There are few published case studies where geostatis-tics is
applied in archaeological contexts. There have,however, been
reviews of geostatistics in archaeology:Ebert [13] and Wheatley and
Gillings [39] both providesummaries of the basic tools of
geostatistics in archaeo-logical contexts. The present paper is
intended to take abroader overview and to outline some existing
applica-tions of geostatistics in archaeology as well as to
presenttwo case studies that are concerned with the analysis of(i)
Roman pottery and (ii) soil phosphate data.First, some published
applications of geostatistics inarchaeology are outlined. Then,
geostatistical theory isintroduced.
* Corresponding author. Tel.: +44-28-8027-3478;fax:
+44-28-9032-1280.
E-mail address: [email protected] (C.D. Lloyd).
Journal of Archaeological Science 31 (2004) 151–165
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2. Published applications of geostatistics in archaeology
In this section, a small number of published studiesare
discussed. These illustrate the wide range of archaeo-logical
problems which geostatistics may help to solve.
Zubrow and Harbaugh [42] is one of only a fewpublications that
apply the geostatistical spatial predic-tion method of kriging in
an archaeological context.Kriging was utilised to reduce the effort
expended inlocating archaeological sites. The sites were locatedin
the archaeological zone of Cañada del Alfaro inGuanajuato, Mexico
and the Hay Hollow valley ineast-central Arizona, USA. The specific
aim of the paperwas to predict, from a sample of the sites
identifiedthrough fieldwork, the expected number of sites in
eachcell of a regular grid. The paper examined the use of asample
of the total surveyed area from which krigedpredictions were made.
The subsequent surveyingrequired to locate all sites in the
surveyed area was thenassessed. It was observed that increasing the
initialsample from 12.5% of the surveyed area to 50% maderelatively
little difference in the number of sites foundin cells predicted by
kriging. In other words, krigingenabled the location of almost as
many of the total sitesfrom 12.5% of the total sample as it did
from 50% of thetotal sample. Thus, in this study, spatial
dependencein the density of sites was demonstrated, as was
theapplicability of methods that utilise this property.
Webster and Burgess [36] examined the application ofkriging to
mapping electrical resistivity for a Saxon orNorman to 17th century
site at Bekesbourne in Kent,England. The data set was used to
illustrate how largescale trends (that is, a spatially varying
mean) may affectthe predictions made using kriging, so the
objective wasonly indirectly archaeological in nature. In a
specificallyarchaeological application, Neiman [27] used
vari-ograms to explore spatial variation in the terminal datesof
Maya settlements (Whitley [40] and Kvamme [20] hada similar
focus).
Geostatistics has been applied in disciplines allied
toarchaeology. Oliver et al. [29] estimated variograms ofleading
principal components and canonical variates ofpollen counts in a
vertical core made through peat inFife, Scotland. Their objective
was to use a range oftools, including variograms (a means of
characterisingspatial structure; defined below), to explore the
structureof the core. Bocquet-Appel and Demars [4]
estimatedvariograms of 14C dates of remains from or associatedwith
European Neanderthals and early modern humans.Models fitted to the
variograms were used to generatemaps representing the spatial
distribution of remains ofdifferent dates.
Robinson and Zubrow [34] discuss interpolation inarchaeology and
they include discussion about kriging,although they caution that
the technique should be usedwith care and that simpler approaches
may be suitable in
many contexts. Hageman and Bennett [15] provide ashort summary
of widely used variants of krigingfor generating Digital Elevation
Models (DEMs) inarchaeological applications.
Ebert [13], in a review of geostatistics for the analysisof
archaeological fieldwalking data, presents an analysisof the
spatial distribution of bulk struck flint. In thatapplication,
cross validation (this entails removing adata point, predicting its
value, comparing the predictedand observed data points and carrying
out the sameprocess for all data) was used to assess the accuracy
ofkriging predictions. A map was also generated using thevariogram
model specified in the paper. Wheatley andGillings [39], in their
review of GIS in archaeology,provide a chapter on interpolation
which includes asection on geostatistical methods. The examples
givenare based on elevation data (as is the focus of Hagemanand
Bennett [15]) and not explicitly archaeological data.
3. Geostatistics
The basic principles of geostatistics are outlinedbelow. There
are many introductions to the subject andseveral more detailed
texts that could be consulted formore information (for example,
[2,14,38]). Burroughand McDonnell [7] provide a short introduction
togeostatistics in the context of GIS. There are alsointroductions
for specific audiences including users ofGISystems [28]; physical
geographers [30,31] and theremote sensing community [9].
3.1. The theory of regionalised variables
In the Earth sciences knowledge about how proper-ties vary in
space is usually sparse. Therefore, it is notfeasible, in general,
to use a deterministic model todescribe spatial variation. If, for
example, the objectiveis to make predictions at locations for which
there are nodata it is necessary to allow for uncertainty in
ourdescription as a result of our lack of knowledge.
The uncertainty inherent in predictions of anyproperty means
that what cannot be described deter-ministically can be accounted
for through the use ofprobabilistic models. With this approach, the
data areconsidered as the outcome of a random process. Isaaksand
Srivastava [17] caution that use of a probabilisticmodel is an
admission of ignorance; it does not meanthat any spatially
referenced property varies randomlyin reality.
In geostatistics, spatial variation (at a location, x)
ismodelled as comprising two distinct parts, a deter-ministic
component (µ(x)) and a stochastic (or ‘random’)component
(R(x)):
Z�x����x��R�x� (1)
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165152
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This is termed a random function (RF) model. Theupper case Z
refers to the RF whereas lower case z refersto the observed data.
In geostatistics, a spatially refer-enced variable, z(x), is
treated as an outcome of a RF,Z(x), defined as a spatial set of
random variables (RVs).A realisation of a RF is called a
regionalised variable(ReV). The Theory of Regionalised Variables
[22] is thefundamental framework on which geostatistics is
based.
Where the properties of the variable of interest are thesame, or
at least similar in some sense, across the regionof interest we can
employ what is termed a stationarymodel. In other words, we can use
the same modelparameters at all locations. Stationarity may be
divided(for geostatistical purposes) into three classes for
whichdifferent parameters of the RF may exist. In turn theseare:
(i) strict stationarity, (ii) second-order stationarityand (iii)
intrinsic stationarity [19,26]. Only the latter twoconcern us
here.
For second-order stationarity, the mean and (spatial)covariance,
are required to be constant. Therefore, theexpected value should be
the same at all locations, x:
E�Z�x���� for all x (2)
In addition, the covariance, C(h), between the locationsx and
x+h should depend only on the lag, h (the dis-tance and direction
by which paired observations areseparated), and not on the
location, x:
C�h��E��Z�x�����Z�x�h�����E�Z�x�Z�x�h���2 for all x (3)
In some cases, the requirements for second-orderstationarity are
not met. For example, the variance (ordispersion) may be unlimited
as lag increases. For thisreason, Matheron [22] defined the
intrinsic hypothesis.For a RF to fulfil the intrinsic hypothesis it
is requiredonly that the expected value of the variable should
notdepend on x:
E�Z�x���� for all x (4)
for all x and the variance of the increments should befinite
[19]. Thus, the variogram, �(h), defined as half theexpected
squared difference between paired RFs, existsand depends only on
h:
�(h)�1
2E[{Z(x)�Z(x�h)}2] (5)
That is, the expected semivariance is the same for
allobservations separated by a particular lag irrespective ofwhere
the paired observations are located.
Second-order stationarity implies the intrinsichypothesis, but
the intrinsic hypothesis does not implysecond-order stationarity.
Thus, the covariance function
and the correlogram (or autocorrelation function,
thestandardised covariance) exist only if the RF is second-order
stationary, and the variogram must be used whenintrinsic
stationarity only can be assumed [19].
3.2. The variogram
The core tool in geostatistical analysis is the vari-ogram
(defined above). The variogram characterisesspatial dependence in
the property of interest. Theexperimental variogram, �̂�h�, can be
estimated from p(h)paired observations, z(xa), z(xa+h), �=1, 2, .
p(h) using:
�̂(h)�1
2p(h)
��1
p(h)
�z(x�)�z(x��h)�2 (6)
In simple terms, the variogram is estimated by calculat-ing the
squared differences between all the availablepaired observations
and obtaining half the average forall observations separated by
that lag (or within a lagtolerance where the observations are not
on a regulargrid). Fig. 1 gives a simple example of a transect
alongwhich observations have been made at regular inter-vals. Lags
(h) of 1 and 2 are indicated. Thus, halfthe average squared
difference between observationsseparated by a lag of 1 is
calculated and the processis repeated for a lag of 2 and so on. The
variogramcan be estimated for different directions to enablethe
identification of directional variation (termedanisotropy).
A mathematical model may be fitted to the exper-imental
variogram and the coefficients of this model canbe used for a range
of geostatistical operations such asspatial prediction (kriging)
and conditional simulation(defined below). A model is usually
selected from one ofa set of authorised models. McBratney and
Webster [24]provide a review of some of the most widely
usedauthorised models. Further models can be found in arange of
texts (for example, [8]).
There are two principal classes of variogram model.Transitive
(bounded) models have a sill (finite variance),and indicate a
second order stationary process (asdefined above). Unbounded models
do not reach anupper bound; they are intrinsic only [24]. Fig. 2
showsthe parameters of a bounded variogram model. Thenugget effect,
c0, represents unresolved variation (a
Fig. 1. Observations (+) made along a transect, with lag (h) of
1 and 2indicated.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 153
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mixture of spatial variation at a finer scale than thesample
spacing and measurement error). The structuredcomponent, c1,
represents the spatially correlated vari-ation. The sill, c0+c1, is
the a priori variance. The range,a, represents the scale (or
frequency) of spatial variation.For example, if soil phosphate
measured at an archaeo-logical site varies markedly over quite
small distancesthen the soil phosphate can be said to have a
highfrequency of spatial variation (a short range) while if thesoil
phosphate is quite similar over much of the site andvaries markedly
only at the extremes of the site (that is,at large separation
distances) then the soil phosphatecan be said to have a low
frequency of spatial variation(a long range).
Variograms used in the case studies presented follow-ing this
section were fitted with a nugget effect and aspherical component.
The nugget variance is given as:
�(h)�H0 if h�01 otherwise (7)The spherical model, a bounded
model, is defined as:
�(h)�5c·[1.5h
a�0.5Sha D
3
] if h#a
c if h>a
(8)
where c is the structured component. Authorised modelsmay be
used in positive linear combination where asingle model is
insufficient to represent well the form ofthe variogram.
3.3. Kriging
There are many varieties of kriging. Its simplest formis called
simple kriging (SK). To use SK it is necessary toknow the mean of
the property of interest and this mustbe modelled as constant
across the region of interest. In
practice this is rarely the case. The most widely usedvariant of
kriging, ordinary kriging (OK), allows themean to vary spatially:
the mean is estimated for eachprediction neighbourhood. OK
predictions are weightedaverages of the n available data. The OK
weights definethe Best Linear Unbiased Predictor (BLUP). The
OKprediction, ẑOK(x0), is defined as:
ẑOK(x0)�
��1
n
��OKz(x�) (9)
with the constraint that the weights, ��OK, sum to 1 to
ensure an unbiased prediction:
��1
n
��OK�1 (10)
So, the objective of the kriging system is to find appro-priate
weights by which the available observations willbe multiplied
before summing them to obtain the pre-dicted value. These weights
are determined using thecoefficients of a model fitted to the
variogram (oranother function such as the covariance function).
The kriging prediction error must have an expectedvalue of
0:
E{ẐOK(x0)�Z(x0)}�0 (11)
The kriging (or prediction) variance, �OK2 , is expressed
as:
�̂OK2 (x0)�E[{ẐOK(x0)�Z(x0)}
2]
���(0)�
��1
n
��1
n
��OK��
OK�(x��x�)�2
��1
n
��OK�(x��x0) (12)
That is, we seek the values of �1, ., �n (the weights)
thatminimise this expression with the constraint that theweights
sum to one (equation 10). This minimisation isachieved through
Lagrange multipliers. The con-ditions for the minimisation are
given by the OK systemcomprising n+1 equations and n+1
unknowns:
5��1n
��OK�(x��x�)��OK��(x��x0) ��1,...,n
��1
n
��OK�1
(13)
where �OK is a Lagrange muliplier. Knowing �OK, theprediction
variance of OK can be given as:
�̂OK2 ��OK��(0)�
��1
n
��OK�(x��x0) (14)
range (a)
nugget (c )
sill (c c )
0
Lag(h)
structured component (c )1
0 1+
Fig. 2. The parameters of a bounded variogram model with a
nuggeteffect.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165154
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The kriging variance is a measure of confidence inpredictions
and is a function of the form of the vari-ogram, the sample
configuration and the sample support(the area over which an
observation is made, which maybe approximated as a point or may be
an area) [19]. Thekriging variance is not conditional on the data
valueslocally and this has led some researchers to use alterna-tive
approaches such as conditional simulation (dis-cussed in the next
section) to build models of spatialuncertainty [14].
There are two varieties of OK: punctual OK andblock OK. With
punctual OK the predictions cover thesame area (the support, V) as
the observations. In blockOK, the predictions are made to a larger
supportthan the observations. With punctual OK the data
arehonoured. That is, they are retained in the output map.Block OK
predictions are averages over areas (i.e., thesupport has
increased). Thus, the prediction is not thesame as an observation
(at x0) and does not need tohonour it.
A worked example of the OK system is provided byBurrough and
McDonnell [7], box 6.2).
3.4. Conditional simulation
Kriging predictions are weighted moving averages ofthe available
sample data. Kriging is, therefore, asmoothing interpolator.
Conditional simulation (alsocalled stochastic imaging) is not
subject to the smooth-ing associated with kriging (conceptually,
the variationlost by kriging due to smoothing is added back)
aspredictions are drawn from equally probable joint reali-sations
of the RVs which make up a RF model [11].That is, simulated values
are not the expected values(i.e., the mean) but are values drawn
randomly from theconditional cumulative distribution function
(ccdf): afunction of the available observations and the
modelledspatial variation [12]. The simulation is considered
“con-ditional” if the simulated values honour the observationsat
their locations [11]. Simulated realisations represent apossible
reality whereas kriging does not. Simulationallows the generation
of many different possible realis-ations that may be used as a
guide to potential errors inthe construction of a map [18] and
multiple realisationsencapsulate the uncertainty in spatial
prediction.
Probably the most widely used form of conditionalsimulation is
sequential Gaussian simulation (SGS).With sequential simulation,
simulated values are condi-tional on the original data and
previously simulatedvalues [11]. In SGS the ccdfs are all assumed
to beGaussian.
The SGS algorithm follows several steps [10,14] asdetailed
below:
1. Apply a standard normal transform to the data.2. Go to the
location x1.
3. Use SK (note OK is often used instead; see Deutschand Journel
[11] about this issue), conditional on theoriginal data, z(x�), to
make a prediction. The SKprediction and the kriging variance are
parameters(the mean and variance) of a Gaussian ccdf:
F(x1;z�(n)�Prob{Z(x1)#z�(n)} (15)
4. Using Monte Carlo simulation, draw a randomresidual, zl(x1),
from the ccdf.
5. Add the SK prediction and the residual which givesthe
simulated value; the simulated value is added tothe data set.
6. Visit all locations in random order and predict usingSK
conditional on the n original data and the i�1values, zl(xi),
simulated at the previously visitedlocations xj, j=1, ., i�1 to
model the ccdf:
F(xi;z�(n�i�1)�Prob{Z(xi)#z�(n�i�1)} (16)
7. Follow the procedure in steps 4 and 5 until alllocations have
been visited.
8. Back transform the data values and simulatedvalues.
By using different random number seeds the order ofvisiting
locations is varied and, therefore, multiple reali-sations can be
obtained. In other words, since thesimulated values are added to
the data set, the valuesavailable for use in simulation are partly
dependent onthe locations at which simulations have already
beenmade and, because of this, the values simulated at anyone
location vary as the available data vary.
SGS is discussed in detail in several texts (forexample,
[8,10,11,14]). The use, and benefits, of SGS areexplored in this
paper.
3.5. Sampling design
Kriging predicts with minimum prediction or krigingerror, �OK
(from here on generalised to �K), and alsopredicts this kriging
error for every predicted value. Thekriging error depends only on
the geometry of thedomain or support V to be predicted, the
distancesbetween V and the n(x0) data points x�, the geometry ofthe
n(x0) data, and finally the variogram [19]. The valuesof the sample
observations locally have no influence.Thus, if the variogram is
known, the kriging error can bepredicted for any proposed sampling
strategy prior tothe actual survey. Kriging is, therefore, an ideal
tool fordesigning optimal sampling strategies.
Burgess et al. [6] chose as their criterion of a goodsampling
strategy, the minimisation of the maximumKriging error, �Kmax. The
quantity �Kmax is not constantover the region of interest, but
rather tends to increasethe further the point (or block) to be
predicted is from
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 155
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the observations (at least for monotonic increasing
vari-ograms). The �Kmax is reached when the point to bepredicted is
furthest from the sample observations, at adistance dmax.
Burgess et al. [6] showed that where spatial variationis
isotropic (invariant with orientation) an equilateraltriangular
grid minimises dmax and hence �Kmax. Anyalternative sampling
schemes, and in particular therandom scheme, will have some larger
values of dmax andhence larger values of �Kmax, although a
hexagonal gridmay be optimum in restricted circumstances [41].
Inpractice, a square grid is likely to be preferred forreasons of
convenience in indexing, site location andcomputer handling, and of
shorter travelling distances inthe field.
The optimal sampling density for a given samplingscheme, can be
designed by solving the Kriging equa-tions for several sampling
densities and plotting �̂Kmaxagainst sample spacing [23,25]. If the
budget for thesurvey is limited then so too is the maximum
precisionattainable. If the survey is not limited by funding and
theinvestigators can define a maximum tolerable predictionerror,
then the optimal sampling strategy is the one thatjust achieves the
desired precision. Greater precisionwould be wasteful. The optimal
strategy is found byreading the required sample spacing from the
plot of�̂Kmax against sample spacing.
The above approach for optimal sampling designprovides a
model-based framework for selecting asample spacing to achieve a
desired precision of predic-tion. However, this approach has been
criticised becauseit does not provide an adequate measure of local
uncer-tainty (e.g., [14]). True, the quantity �̂Kmax
generallyincreases with dmax such that densely sampled areas
havesmaller values of �̂Kmax than sparsely sampled areas.However,
the quantity �̂Kmax is not affected by thecharacter of spatial
variation locally. Thus, in terms ofelevation, mountainous regions
and floodplain areaswould result in the same �̂Kmax, for a given
samplingframework. This inadequacy is most evident in maps of�̂Kmax
for gridded data: the same local pattern in �̂Kmax isrepeated
globally. Despite these limitations �̂Kmax can beuseful as a guide
to uncertainty in predictions wherespatial variation is similar
across the region of interest.In cases where the form of spatial
variation changesacross the region of interest a non-stationary
approach(for example, splitting the data into sub-sets which canbe
regarded as ‘homogeneous’) can be applied [21].
3.6. Software
The wide range of public domain and low costsoftware now
available (see [35], for a review of somepublic domain software)
means that the tools of geo-statistics are readily available to the
archaeologist.Widely used public domain software packages
include
GSLIB (Geostatistical Software Library, [11] and Gstat[32], both
used for the case studies presented in thispaper. In addition,
several commercial GISystemsinclude geostatistical functions and
there is a range ofcommercial geostatistical packages.
4. Case studies
In this section, two case studies are presented. Thefirst case
study is an analysis of the distribution ofRoman pottery in
southern Britain and use of thevariogram is illustrated. The second
case study showshow the variogram, kriging (punctual and block
OK)and conditional simulation (SGS) can be applied to theanalysis
of the distribution of soil phosphates at anarchaeological site in
Greece.
4.1. Case study 1: Roman pottery in southern Britain
The first case study utilises the variogram to charac-terise
spatial dependence in assemblages of Romanpottery from the south of
Britain from details collectedby Allen and Fulford [1]. Allen and
Fulford acquireddata on five types of pottery, but of these, only
twooccur with enough regularity at the sites surveyed toprovide a
large enough sample for geostatistical analysis.The two types
considered here are South-East DorsetBlack Burnished Category I
(SEDBB I) and SevernValley Ware (SVW). SVW was not recorded at many
ofthe sites and variograms estimated from few data areoften ‘noisy’
and visually unstructured. It should also benoted that the
percentages of SEDBB I and SVW ateach site were obtained in various
different ways includ-ing sherd counts, sherd weights, estimated
vessel equiva-lent (EVE) and number of vessels represented
(VR).Allen and Fulford [1] discuss this issue in some detail.
The omnidirectional variogram for SEDBB I is pre-sented in Fig.
3. The increase in semivariance with lagfor the variogram of SEDBB
I percentages is indicativeof spatial dependence and a model was
fitted to thisvariogram. There is a clear tendency for semivariance
toincrease up to a lag of about 90 km after whichsemivariance
remains constant. The range (a) of thefitted variogram model was
119.91 km. This may beinterpreted as the separation distance above
whichassemblages of SEDBB I are spatially independent.
Inarchaeological terms, this may represent the redistribu-tion of
pottery of this type from production centres tomarkets. In other
words, pottery types that exhibitclearly structured spatial
variation may be consideredexamples of larger scale production,
vessels that perhapsdominate in the region of concern and are
foundconsistently in archaeological assemblages. In such ascenario,
industries that were only local in scale would,in a regional
context, be marked by unstructured spatialvariation. A map of SEDBB
I%, derived using OK, is
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165156
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given in Fig. 4 (the validity of mapping properties suchas
artefact proportions is discussed below). Predictionsare shown only
within 250 km of the observations. Thelargest SEDBBI concentrations
are in Dorset, as wouldbe expected. The linear features visible in
the map arecharacteristic of areas located far away from
sampledata.
In this case, the map should not be viewed as a mapof predicted
pottery amounts (that is, percentages) sincepottery amount is not a
continuously varying property(unlike, for example, elevation or
rainfall). As Wheatleyand Gillings [39] note, surfaces derived from
observa-tions such as counts of artefacts may be useful where
thesurvey from which the data derive was not exhaustive (togive an
indication of counts at areas where no data areavailable) but such
maps should be viewed with caution.The kriged surface provides a
way of gaining a clearersense of regional variation in SEDBB I
amount than ispossible using shaded point maps and it may be
con-sidered to represent the idealised catchment of SEDBB I,but it
does not represent pottery amount (that is,percentage) per se.
The omnidirectional variogram for SVW in Fig. 5appears to
demonstrate no general increase in semivari-ance with increase in
lag h—in such a case a nugget effectonly may describe adequately
the form of the variogramand this would be referred to as pure
nugget. In otherwords, semivariance does not increase markedly
withrespect to the nugget variance as lag increases. Direc-tional
variograms were also computed and, in mostcases, gave little
indication of spatial dependence. How-ever, the variogram for 0(
(north-south alignment) givenin Fig. 6, to which a model was
fitted, demonstrates afairly clear increase in semivariance as lag
h increases.
This indicates that the distribution of SVW is morecontinuous in
the north–south direction than in otherdirections. Allen and
Fulford’s contour map of SVWdepicts major contours aligned
north–south with morevisually erratic changes in the contours in an
east-westalignment. This corroborates the form of the vari-ograms.
Additionally, there are more data within thenorth and south extents
of the data set than there arewithin the east–west limits, which
means that the vari-ogram would be most stable in form for the
north–southdirection. However, the variogram is unbounded andthis
may be indicative of differences between the northand south rather
than within-region differences. Thevariogram provides a means to
quantify spatial vari-ation and compare different properties in a
mannerthat is more objective than comparing visually maps.
4.2. Case study 2: Mapping soil phosphates
Kriging has been applied widely in soil survey to mapsoil types
(for example, [37]) and is here used to mapsoil phosphates from an
archaeological site. The dataexamined were published by Buck et al.
[5]. The datarepresent soil phosphate measured at a site
(referenceLS 165), probably dating to the Roman period, that
wasstudied as part of the Laconia Survey in Greece. Themeasures are
mg P/100 g of soil and were obtained at10 m intervals on a 16 by 16
point grid (although nodata were obtained at nine locations on the
grid due toobstacles at those nodes of the grid).
Variograms were estimated and models fitted tothem using Gstat.
The omnidirectional variogram ofsoil phosphate (Fig. 7) illustrates
that the soilphosphate is spatially correlated. The
omnidirectional
0
100
200
300
400
500
600
700
800
900
0 20 40 60 80 100 120 140
Sem
ivar
ianc
e (S
ED
BB
I %
2 )
Lag (km)
Semivariance256.785 Nug(0) + 527.289 Sph(119.91)
Fig. 3. Omnidirectional variogram for SEDBB I. Nug. is nugget,
Sph. is spherical.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 157
-
variogram was fitted with a nugget effect and a sphericalmodel.
The large nugget effect is indicative of uncer-tainty in the
measurement of soil phosphates and local(small-scale) variation in
soil phosphates. The modelfitted to the omndirectional variogram
has a range of80.325 m. This can be interpreted as the maximum
scaleof spatial variation in soil phosphate in this region.
The variogram was also estimated for different direc-tions
within a tolerance of 45 degrees. The variogramwith the largest
range was for 22(:30# (modelled rangeof 80.976 m) while the
variogram at 90( from this had amodelled range of 77.079 m. The
variograms, andstructured components, for those two directions
aregiven in Fig. 8. The models fitted for the two directionshave
similar ranges but different nugget variances.Where the sill
(recall the total sill is the nugget varianceplus the structured
components) differs for differentdirections this is termed zonal
anisotropy. The differ-ences are not marked and the most
straightforwardapproach, to use the coefficients from the omni-
directional variogram model as input for kriging, wasaccepted in
this case.
The OK functionality of GSLIB was used to Krigesoil phosphate to
a grid with a 2 m spacing. Bothpunctual OK and block OK were
applied: the choice ofone of the two approaches is an important
issue. Themap of punctual OK predictions is given in Fig. 9 andthe
corresponding kriging variance in given in Fig. 10.The locations of
the observations are obviousin both Fig. 9 and Fig. 10. In Fig. 9,
the observationsappear as ‘spikes’ in the map. This is a common
featureof maps derived using punctual kriging. The OK vari-ance at
the observation locations is zero in Fig. 10. Thisimplies that
there is no measurement error in the data,but in fact measurement
of soil phosphate entails muchuncertainty. The analysis was
repeated using block OK.The spikes evident in Fig. 9 are not
apparent in the mapof block OK predictions (Fig. 11). Also, the
block OKvariances (Fig. 12), unlike in Fig. 10, are not zero at
anylocations. Note also that the range of values in the block
SEDBB I %
Value
High : 68.90
Low : 2.78
±
0 120,000 240,00060,000 Metres
Fig. 4. Map of SEDBB I%, derived using OK. 1000 m cells.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165158
-
OK map is smaller than the range of values for thepunctual OK
map because of the process of averagingover a 2 m by 2 m block. It
is, as noted previously,important to consider issues such as the
support overwhich predictions will be made.
Buck et al. [5] aimed to delineate areas with high andlow
concentrations of soil phosphate. Although the aimhere has been
simply to demonstrate the applicationof OK for interpolation, other
kriging algorithms, inparticular, disjunctive kriging [33], may be
used to
assess the probability that a predicted value exceeds
aparticular threshold.
Conditional simulation was also applied to the data.Four maps
derived using SGS (the algorithm in GSLIBwas utilised) are given in
Fig. 13. Differences betweenthe four realisations are apparent.
Conditional simula-tion provides a powerful means to explore
variation inspatial data and there are extensive potential
applica-tions for interpreting and mapping distributions
ofarchaeological variables. Each one of the maps in
0
100
200
300
400
500
0 20 40 60 80 100 120 140
Sem
ivar
ianc
e (S
VW
%2 )
Lag (km)
Semivariance
Fig. 5. Omnidirectional variogram for SVW.
0
100
200
300
400
500
600
0 20 40 60 80 100 120 140
Sem
ivar
ianc
e (S
VW
%2 )
Lag (km)
Semivariance290 Nug(0) + 2 Pow(1)
Fig. 6. Directional variogram (0() for SVW. Nug. is nugget, Pow.
is power.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 159
-
Fig. 13 represents a possible reality, whereas neitherFig. 9 or
Fig. 11 is a possible reality because they aresmoothed
representations. Kriging provides the bestprediction on a
point-by-point basis, whereas simulationis the best on a global
basis that is, reproduces theoriginal spatial structure. Statistics
estimated frommultiple simulated realisations may be a useful guide
tospatial uncertainty.
It was noted above that the coefficients of the modelfitted to
the variogram have been used to ascertain the
maximum punctual kriging variance for different samplespacings
(for a prediction neighbourhood of 16 obser-vations). This enables
the researcher to ascertain themaximum sample spacing possible to
achieve a particu-lar precision [3]. To do this it is necessary to
obtain asample data set for which a representative variogrammay be
estimated. Measurements are often made alonga transect for this
purpose. The coefficients of the modelfitted to the omnidirectional
variogram of soil phos-phate were input into the Fortran program
OSSFIM
0
100
200
300
400
500
600
700
800
0 10 20 30 40 50 60 70 80
Sem
ivar
ianc
e (m
g P
/100
g)2
Lag (m)
Semivariance409.959 Nug(0) + 411.119 Sph(80.325)
Fig. 7. Omnidirectional variogram of soil phosphate. Nug. is
nugget, Sph. is spherical, DD is decimal degrees.
0
100
200
300
400
500
600
700
800
900
0 10 20 30 40 50 60 70 80
Sem
ivar
ianc
e (m
g P
/100
g)2
Lag (m)
Semivariance: 22.5 dd22.5 DD: 443.898 Nug(0) + 410.956
Sph(80.976)
Semivariance: 112.5 dd112.5 DD: 366.037 Nug(0) + 417.243
Sph(77.079)
Fig. 8. Directional variogram of soil phosphate for 22:30( and
112:30(. Nug. is nugget, Sph. is spherical, DD is decimal
degrees.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165160
-
(Optimal Sampling Schemes for Isarithmic Mapping,[23,25] and the
maximum kriging variance, �Kmax, forseveral different sample
spacings was obtained (Fig. 14).In Fig. 15, it is shown that if a
required �Kmaxof 625 mg
P/100 g2 (that is, 25 mg P/100 g) were stated then asample
spacing of about 25 m would be necessary. Thekriging variance is
directly dependent on the form of thevariogram so it is necessary
that the variogram is
0 m 100 m0 m
100 m
27.000
37.000
47.000
57.000
67.000
77.000
87.000
97.000
107.000
117.000
127.000
137.000±
Fig. 9. Map of soil phosphate produced using punctual OK, 2 m
cells. Scale is in mg P/100 g of soil.
0 m 100 m0 m
100 m
0.0
71.068
142.136
213.204
284.272
355.340
426.408
497.476
568.544
639.612
710.680±
Fig. 10. Map of punctual OK variances. Scale is in (mg P/100 g)2
of soil.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 161
-
representative of the region for which it is estimated.If this
is the case, this approach could be a useful toolfor the
archaeologist as an aid to designing samplingstrategies.
5. Summary and conclusions
Geostatistics offers many potential benefits to archae-ologists
who are concerned with the analysis of spatial
0 m 100 m0 m
100 m
44.000
54.000
64.000
74.000
84.000
94.000
104.000±
Fig. 11. Map of soil phosphate produced using block OK, 2 m
cells. Scale is in mg P/100 g of soil.
0 m 100 m0 m
100 m
81.100
102.418
123.736
145.054
166.372
187.690
209.008
230.326
251.644
272.962
294.280±
Fig. 12. Map of block OK variances. Scale is in (mg P/100 g)2 of
soil.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165162
-
data. However, like any tool geostatistics must be
usedappropriately. In many cases, simpler tools may beappropriate.
So, it is necessary to consider carefully thepros and cons of
geostatistics in any given situation.
Where the spatial variation in an archaeologicalvariable is of
interest the tools of geostatistics havemuch potential value. Tools
such as the variogram maybe utilised to quantify and interpret
observed spatial
0 m 100 m 0 m
100 m
17.000
57.000
97.000
137.000
177.000
217.000
257.000
297.000
0 m 100 m 0 m
100 m
17.000
57.000
97.000
137.000
177.000
217.000
257.000
297.000
0 m 100 m 0 m
100 m
17.000
57.000
97.000
137.000
177.000
217.000
257.000
297.000
0 m 100 m 0 m
100 m
17.000
57.000
97.000
137.000
177.000
217.000
257.000
297.000
Fig. 13. Four maps of soil phosphate produced using SGS, 2 m
cells. Scale is in mg P/100 g of soil.
550
600
650
700
0 5 10 15 20 25 30 35 40
Max
. krig
ing
var.
(m
g P
/100
g)
Sample spacing (m)
2
Fig. 14. Plot of maximum kriging variance against sample
spacing.
C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 163
-
distributions. Hodder and Orton [16] illustrated howmeasures of
spatial autocorrelation could be used tocharacterise spatial
variation in archaeological variables.In addition to
characterisation of spatial dependence,this paper has demonstrated
how geostatistics may beused to analyse and map archaeological
variables.There are many archaeological variables that could
beanalysed geostatistically. Some obvious ones are
artefactdensities and dates of objects [39]. The
followingapplications were outlined (using the tools specified
inparentheses):
• characterisation of spatial variation (variogram)• spatial
prediction (ordinary kriging)• assessment of uncertainty in mapped
predictions
(kriging variance)• conditional simulation (sequential Gaussian
simula-
tion)• design of optimal sampling strategies (kriging
variance).
The tools of geostatistics represent a powerfuladdition to the
archaeologist’s tool kit but, so far, littleof the potential
benefits have been realised. This is due,in part, to the perceived
complexity of the techniquesand the models that underlie them. It
is hoped that thispaper will serve in some way to expand the
under-standing of geostatistics and to encourage its use
inarchaeology.
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C.D. Lloyd, P.M. Atkinson / Journal of Archaeological Science 31
(2004) 151–165 165
Archaeology and geostatisticsIntroductionPublished applications
of geostatistics in archaeologyGeostatisticsThe theory of
regionalised variablesThe variogramKrigingConditional
simulationSampling designSoftware
Case studiesCase study 1: Roman pottery in southern BritainCase
study 2: Mapping soil phosphates
Summary and conclusions
References